Simplify the following expression: $y = \dfrac{5x^2+4x- 9}{5x + 9}$
Solution: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(5)}{(-9)} &=& -45 \\ {a} + {b} &=& &=& {4} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-45$ and add them together. Remember, since $-45$ is negative, one of the factors must be negative. The factors that add up to ${4}$ will be your ${a}$ and ${b}$ When ${a}$ is ${9}$ and ${b}$ is ${-5}$ $ \begin{eqnarray} {ab} &=& ({9})({-5}) &=& -45 \\ {a} + {b} &=& {9} + {-5} &=& 4 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({5}x^2 +{9}x) + ({-5}x {-9}) $ Factor out the common factors: $ x(5x + 9) - 1(5x + 9)$ Now factor out $(5x + 9)$ $ (5x + 9)(x - 1)$ The original expression can therefore be written: $ \dfrac{(5x + 9)(x - 1)}{5x + 9}$ We are dividing by $5x + 9$ , so $5x + 9 \neq 0$ Therefore, $x \neq -\frac{9}{5}$ This leaves us with $x - 1; x \neq -\frac{9}{5}$.